Suppose an integer from 1 through 1000 is chosen at random, fins the probability that the integer is a multiple of 2 or a multiple of 9.
Given, Sample space is the set of first 1000 natural numbers.
∴ n(S) = 1000
Let A be the event of choosing the number such that it is multiple of 2
∴ n(A) = [1000/2] = [500] = 500 {where [.] represents Greatest integer function}
∴ P(A) =
Let B be the event of choosing the number such that it is multiple of 9
∴ n(B) = [1000/9] = [111.11] = 111 {where [.] represents Greatest integer function}
∴ P(B) =
We need to find the P(such that number chosen is multiple of 2 or 9)
∵ P(A or B) = P(A ∪ B)
Note: By definition of P(E or F) under axiomatic approach(also called addition theorem) we know that:
P(E ∪ F) = P(E) + P(F) – P(E ∩ F)
∴ P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
We don’t have value of P(A ∩ B) which represents event of choosing a number such that number is a multiple of both 2 and 9 or we can say that it is a multiple of 18.
n(A ∩ B) = [1000/18] = [55.55] = 55
∴ P(A ∩ B) =
∴ P(A ∪ B) =